Numerical Differentiation

such as these arise frequently in scientific applications, generally as a result of the inherent errors present in measurement processes. If one were to attempt to use standard numerical techniques, such as the central difference formula, to compute approximated derivative values for the underlying function, it is evident that the errors in these derivatives would greatly exceed the measurement error present in the original function data. In fact, it has long been known that with no restrictions on the type of uniform perturbation that one allows, one can construct examples in which the difference between the original and perturbed derivative values is arbitrarily large. The real problem from a practical standpoint therefore is this: assuming some kind of uniform perturbation, one seeks to delineate a class of suitable perturbations, with the intention of being able to compute, within the class, an approximate derivative whose magnitude of error is approximately that of the original uniform perturbation. The “suitable class ” here would typically depend on the application, but is generally clear in specific cases.